Theorem oncard 4839

Description: A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85.

Assertion

Ref	Expression
oncard	|- (E.x A = (card` x) <-> A = (card` A))

Distinct variable group:   x,A

Proof of Theorem oncard

Step	Hyp	Ref	Expression
1		cardid 4838	. . . . . . 7 |- (card` x) ~~ x
2		breq1 2627	. . . . . . 7 |- (A = (card` x) -> (A ~~ x <-> (card` x) ~~ x))
3	1, 2	mpbiri 194	. . . . . 6 |- (A = (card` x) -> A ~~ x)
4		cardid 4838	. . . . . . 7 |- (card` A) ~~ A
5		entrt 4420	. . . . . . 7 |- (((card` A) ~~ A /\ A ~~ x) -> (card` A) ~~ x)
6	4, 5	mpan 697	. . . . . 6 |- (A ~~ x -> (card` A) ~~ x)
7		cardon 4837	. . . . . . . 8 |- (card` A) e. On
8		breq1 2627	. . . . . . . . 9 |- (y = (card` A) -> (y ~~ x <-> (card` A) ~~ x))
9	8	onintss 3017	. . . . . . . 8 |- ((card` A) e. On -> ((card` A) ~~ x -> |^|{y e. On | y ~~ x} (_ (card` A)))
10	7, 9	ax-mp 7	. . . . . . 7 |- ((card` A) ~~ x -> |^|{y e. On | y ~~ x} (_ (card` A))
11		cardval 4836	. . . . . . 7 |- (card` x) = |^|{y e. On | y ~~ x}
12	10, 11	syl5ss 2108	. . . . . 6 |- ((card` A) ~~ x -> (card` x) (_ (card` A))
13	3, 6, 12	3syl 20	. . . . 5 |- (A = (card` x) -> (card` x) (_ (card` A))
14		sseq1 2085	. . . . 5 |- (A = (card` x) -> (A (_ (card` A) <-> (card` x) (_ (card` A)))
15	13, 14	mpbird 196	. . . 4 |- (A = (card` x) -> A (_ (card` A))
16		cardon 4837	. . . . . 6 |- (card` x) e. On
17		eleq1 1537	. . . . . 6 |- (A = (card` x) -> (A e. On <-> (card` x) e. On))
18	16, 17	mpbiri 194	. . . . 5 |- (A = (card` x) -> A e. On)
19		cardonle 4832	. . . . 5 |- (A e. On -> (card` A) (_ A)
20	18, 19	syl 10	. . . 4 |- (A = (card` x) -> (card` A) (_ A)
21	15, 20	eqssd 2082	. . 3 |- (A = (card` x) -> A = (card` A))
22	21	19.23aiv 1297	. 2 |- (E.x A = (card` x) -> A = (card` A))
23		fvex 3738	. . . 4 |- (card` A) e. V
24		eleq1 1537	. . . 4 |- (A = (card` A) -> (A e. V <-> (card` A) e. V))
25	23, 24	mpbiri 194	. . 3 |- (A = (card` A) -> A e. V)
26		fveq2 3730	. . . . 5 |- (x = A -> (card` x) = (card` A))
27	26	eqeq2d 1489	. . . 4 |- (x = A -> (A = (card` x) <-> A = (card` A)))
28	27	cla4egv 1866	. . 3 |- (A e. V -> (A = (card` A) -> E.x A = (card` x)))
29	25, 28	mpcom 49	. 2 |- (A = (card` A) -> E.x A = (card` x))
30	22, 29	impbi 157	1 |- (E.x A = (card` x) <-> A = (card` A))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  E.wex 982  {crab 1651  Vcvv 1814   (_ wss 2050  |^|cint 2537   class class class wbr 2624  Oncon0 2954  ` cfv 3188   ~~ cen 4370  cardccrd 4823

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-er 4267  df-en 4374  df-card 4826

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